New approaches in the chemometric analysis of infrared spectra of

Statistics & Operations Research Transactions
SORT 38 (2) July-December 2014, 231-250
ISSN: 1696-2281
eISSN: 2013-8830
www.idescat.cat/sort/
Statistics &
Operations Research
c Institut d’Estadı́stica de Catalunya
Transactions
[email protected]
New approaches in the chemometric analysis
of infrared spectra of extra-virgin olive oils
Marı́a Isabel Sánchez-Rodrı́guez1,∗ , Elena M. Sánchez-López2 ,
Alberto Marinas2 , José Ma Caridad1 , Francisco José Urbano2
and José Ma Marinas2
Abstract
The aim of this paper is to apply new chemometric approaches to obtain quantitative information
from near and mid infrared spectra of Andalusian extra-virgin olive oils, using gas chromatography
as a classical reference analytical technique. Estimations of the content in saturated, monounsaturated and polyunsaturated fatty acids are given using partial least squares regression from the
near and mid infrared data matrices as well as their concatenated matrix. The different estimations
are evaluated in terms of goodness of fit (calibration) and prediction (validation), as a function of
the number of partial least squares factors in the regression model and the used matrix of data.
Furthermore, the nature, systematic or random, of the prediction errors is studied by a decomposition of their mean squared error. Finally, procedures of cross-validation are implemented in
order to generalize the previous results.
MSC: 62H25, 62J05, 62Q99.
Keywords: Extra-virgin olive oil, infrared spectroscopy, partial least squares regression, crossvalidation.
1. Introduction
Extra-virgin olive oil is an edible oil very much appreciated by its taste and benefits for
health. Mediterranean countries (Spain, Italy, Greece, Turkey, Tunisia and Morocco)
and Portugal cover 90% of the world production, Spain and Italy being the major
producers and consumers. In Spain, Andalusia produces 80% of the national product.
∗ Corresponding
author e-mail: [email protected]
Estadı́stica, Econometrı́a, I.O., Org. Empresas y Ec. Aplicada. University of Córdoba
2 Dep. Quı́mica Orgánica. University of Córdoba
Received: September 2013
Accepted: June 2014
1 Dep.
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New approaches in the chemometric analysis of infrared spectra of extra-virgin olive oils
The composition of olive oil depends on the type and the distribution of the fatty acids
present in the triglycerides and on the positions in which these fatty acids are esterified
to hydroxyl groups in glycerol backbone. The principal fatty acids of vegetable oils
are oleic, linoleic, linolenic, myristic, palmitic and estearic. The last three types are
classified as saturated (SAFA), the oleic is monounsaturated (MUFA) and the linoleic
and linolenic acids are polyunsaturated (PUFA).
Extra-virgin oil is by definition obtained only from the olive, using solely mechanical
or other physical means, in conditions, particularly thermal conditions, which do not
alter the oil in any way. It presents a high price of commercialization, which makes
it susceptible to adulteration with other cheaper oils, such as hazelnut, sunflower,
soybean, maize or refined olive oils (see, for example, Baeten et al. (2005), Gurdeniz
and Ozen (2009) and Öztürk et al. (2010)), considerably modifying its quality indices.
This makes it necessary to provide fast, reliable and cost-effective analytical procedures
which require no or little sample manipulation. In this sense, for several years we have
elaborated an extra-virgin olive oils database using diverse spectroscopic techniques
such as near and mid infrared (NIR and MIR, respectively). IR techniques provide
continuous information (spectra) that is rich in both isolated and overlapping bands
and not so obvious to analyse as in the case of gas chromatography (GC). Nevertheless,
the application of multivariate statistics to the above-mentioned spectra allows to obtain
quantitative information (as the content of oil in diverse compounds) or qualitative (as
the geographical origin or the protected designation of origin, PDO) about the olive oil.
There are in the literature diverse examples of application of NIR, MIR or concatenated NIR-MIR spectroscopic techniques to the quantitative and qualitative analysis of
olive oils. Thus, for example, Bertran et al. (2000) and Galtier et al. (2007, 2011) classify several olive oils according to different geographical zones and determine the composition in fatty acids and triacylglycerols by using NIR spectra. Baeten et al. (2005)
and Gurdeniz and Ozen (2009) study the possible adulteration of olive oils with lower
quality oils (such as hazelnut, sunflower or maize) by MIR spectroscopy. Dupuy et al.
(2010a, 2010b) and Sinelli et al. (2008) use NIR, MIR and concatenated NIR-MIR spectra to develop quantitative and qualitative studies of olive oil. Sinelli et al. (2010) apply
NIR and MIR spectroscopies as a rapid tool to classify extra virgin olive oil on the basis
of fruity attribute intensity. Casale et al. (2012) use NIR and MIR spectroscopical data,
individually and jointly, to characterize olive oils from an Italian protected designation
of origin.
Some of these works determine, using correlation, specially significant IR spectral
bands to fit regression models to predict some components of olive oil (see Guillén and
Cabo (1997) or Zhang et al. (2012)). Other works, such as Maggio et al. (2011), use
partial least squares (PLS) regression to avoid the presence of multicollinearity in the
model. PLS regression summarizes the information of a spectral band in some components or latent factors being orthogonal among them and so avoiding multicollinearity,
incompatible with the hypothesis of uncorrelation in the general linear model. Other
authors, such as Casale et al. (2012) or Dupuy et al. (2010a, 2010b), extract the PLS
Marı́a Isabel Sánchez-Rodrı́guez et al.
233
components from the complete NIR, MIR or concatenated NIR-MIR spectra, not from
the previously selected spectral bands.
The aims of this paper are to revisit the procedures used in the literature to obtain
quantitative information of olive oil from the near and mid zones of the infrared spectra
and propose new approaches. The goal is to determine the profile in SAFA, MUFA and
PUFA fatty acids of diverse extra-virgin olive oils by using the information provided by
the NIR, MIR and concatenated NIR-MIR matrices of data, using the values obtained
from GC as a reference . The estimations are provided by partial least squares regression
models and are compared in terms of goodness of fit (calibration) and prediction
(validation), that is, measuring errors that correspond to data used or not used to train the
regression model. In addition, a decomposition of the mean squared error of prediction
is provided to evaluate the character, systematic or random, of prediction errors (see
Sánchez-Rodrı́guez et al. (2013)) . The obtained results are generalized using procedures
of cross-validation, based on the design of repetitive algorithms that, for each iteration,
modify the partition of the available data set in subsets of calibration and validation.
Finally, three-dimensional scatterplots give a global vision for the three types of fatty
acids and matrices of data, simultaneously.
The computer programs commonly used in Chemometrics have internally implemented a stopping criterion to determine the number of PLS latent factors to retain in
the regression model. But, in this work, the PLS factors are progressively introduced
in the model, with the aim of determining the evolution of calibration and validation
errors as a function of the number of factors and the estimated fatty acid type. The
chemometric software has also cross-validation procedures implemented that change,
at each iteration, the learning and the test data sets, and provide a global mean of the
calibration and validation errors. On the contrary, in the present work, the procedures
of cross-validation have been programmed and show the results corresponding to each
iteration. Therefore, the evolution and the variability of the fit and prediction errors can
be studied for the successive iterations.
2. Acquisition of data
The studied samples include 128 Andalusian extra-virgin olive oils, collected for four
consecutive seasons (from 2007 to 2011) with a ripeness index of 3. The varieties studied
are, mainly, ‘Arbequina’, ‘Hojiblanca’, ‘Picual’, ‘Lechı́n’, ‘Manzanilla’, ‘Picudo’ and
‘Royal’. Olive oil was extracted by the producers through a two-phase centrifugation
system. The data for the subsequent statistical treatment have been provided by the
following analytical chemical procedures:
• Gas chromatography. Classical separation technique that leads to discrete information including several usually well-defined, separated peaks from which,
on proper integration, the content of various chemical components (for example,
234
New approaches in the chemometric analysis of infrared spectra of extra-virgin olive oils
SAFA, MUFA and PUFA fatty acids) can be determined. It will be considered as
reference technique in the next studies.
• Spectroscopical techniques. Infrared techniques, such as NIR and MIR, generate
continuous information, rich in both isolated and overlapping bands attributed to
vibration of chemical bonds in different molecules. The use of mathematical and
statistical procedures allows us to extract the maximum useful information from
data (Berrueta et al. (2007)).
2.1. Gas Chromatography (GC)
The determinations of fatty acid composition by GC-FID, according to the official methods for olive and pomace oil control in the European Union, EU (2011) and the International Olive Council, COI (2001a, 2001b), were performed by the staff of Organic
Chemistry of University of Córdoba, using an Agilent 7890A gas chromatograph with a
capillary column (SGE FORTE BPX-70 de 50 m × 220 µm × 0.25 µm). The conditions
of analysis were as follows: 250 ◦ C of injector temperature, 2 µL of injection volume,
260 ◦ C of detector temperature. The oven temperature was programmed to remain at
180 ◦ C for 15 min and then raised to 240 ◦ C at a rate of 4 ◦ C/min and maintained at this
temperature for 5 min.
The triacylglycerol samples (olive oil samples), were initially submitted to a cold
transesterefication procedure to convert the triacylglycerol into fatty acid methyl esters.
This method is indicated for edible oils with acidity index lower than 3.3◦ . In this
process, 0.1 g of olive oil are transferred into a 5 mL volumetric flask. Next, 2 mL
n-heptane and 0,2 mL of a 2N KOH solution in methanol were added and reaction
mixture was vigorously stirred. Finally, the methyl esters were extracted and subject to
GC analyses.
2.2. NIR and MIR spectra
NIR and MIR spectra were obtained by the staff of the Organic Chemistry Department
of the University of Córdoba within 15 days after reception of the samples which where
kept in the fridge so that properties were not modified [Baeten et al. (2003)]. The
instruments employed for spectra collection were available at the Central Service of
Analyses (SCAI) at the University of Córdoba.
As for NIR instrument, it consisted in a Spectrum One NTS FT-NIR spectrophotometer (Perkin Elmer LLC, Shelton, USA) equipped with an integrating sphere module. Samples were analyzed by transflectance by using a glass petri dish and a hexagonal
reflector with a total transflectance pathlength of approximately 0.5 mm. A diffuse reflecting stainless steel surface placed at the bottom of the cup reflected the radiation
back through the sample to the reflectance detector. The spectra were collected by us-
Marı́a Isabel Sánchez-Rodrı́guez et al.
Figure 1: NIR spectrum of an extra-virgin
olive oil.
235
Figure 2: MIR spectrum of an extra-virgin
olive oil.
ing Spectrum Software 5.0.1 (Perkin Elmer LLC, Shelton, USA). The reflectance
(log 1/R) spectra were collected with two different reflectors. Data correspond to the
average of results with both reflectors, thus ruling out the influence of them on variability of the obtained results. Moreover, spectra were subsequently smoothed using the
Savitzky-Golay technique, which performs a local polynomial least squares regression
in order to reduce the random noise of the instrumental signal. Once pre-treated, NIR
data of 1237 measurements for each case (representing energy absorbed by olive oil
sample at 1237 different wavelengths, from 800.62 to 2499.64 nm) were supplied to the
Department of Statistics (University of Córdoba) in order to be analysed.
Regarding MIR spectra of olive oil samples, they contain both well-resolved (31001721 cm−1 ) and overlapping peaks (1500-700 cm−1 ). Spectra were registered at room
temperature in the 600 to 4000 cm−1 range on a Tensor 27 FTIR Spectrometer (Bruker
Optics, Milano, Italy) coupled to an ATR (Attenuated total reflectance) device consisting
in several reflection crystals (ZnSe). Software used was OPUS r. 5,0 (Bruker Optics), the
resolution 2 cm−1 and 50 scan per sample. The number of measurements for each case
was 1843, which were supplied to the Department of Statistics (University of Córdoba)
for analysis.
3. Multivariate data analysis
3.1. Selection criteria of regression models
The purpose of this work is to use statistical regression models to determine the profile
in fatty acids SAFA, MUFA and PUFA of extra-virgin olive oils obtained by gas
chromatography (classical technique used as reference) from the information provided
by the IR spectroscopy technique. The regression models are evaluated in terms of
goodness-of-fit and predictive capability, using the following measures.
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New approaches in the chemometric analysis of infrared spectra of extra-virgin olive oils
Let y1 , y2 , . . . , yn be the observations of a dependent variable, Y , and the corresponding predictions, yb1 , yb2 , . . . , ybn , of a regression model. The mean squared error of calibration, MSEC= ∑ni=1 (yi − ybi )2 /n, takes values nearer to 0 for a good fit (calibration). But
MSEC is not dimensionless, that is, it depends on the units of measurement of the variable, hence it is useful. It is useful in comparisons of models with variables measured in
the same units but not as an absolute goodness of fit measure.
Given the predictions for the future t observations, ybn+1 , ybn+2 , . . . , ybn+t , of a certain regression model, the mean squared error of the prediction, MSEP= ∑tj=1 (yn+ j − ybn+ j )2 /t,
evaluates the predictive capability (validation) of a regression model. The predictive capability of a model is obviously better as MSEP approaches 0, taking into account that
MSEC has no upper bound and depends on the measurement units.
As indicated by Berrueta et al. (2007), the ideal situation in the evaluation of the
predictive capability of a model is when there are enough data available to create separate test sets completely independent from the model building process (this validation
procedure is known as external validation). When an independent test set is not available (e.g. because cost or time constraints make it difficult to increase the sample size),
MSEP has to be estimated from the data used to train the regression. For this reason, as
validation set, part of the original data set is used, avoiding the bias associated to the fact
that the same data are used to the fit of the regression model and the evaluation of the
predictions). The cross-validation procedures are designed to modify the selections repeatedly, using an algorithm that, for each iteration, changes the partition of the original
data set into calibration and validation sets.
Besides, in line with the approach introduced by Fisher around 1920 relative to
analysis of variance, Sánchez-Rodrı́guez et al. (2013) described new insights into
evaluation of regression models through a decomposition of MSEP to analyse more
in depth the causes of the prediction errors. Let y and yb be the means of the t future
observations and their predictions, sY and sYb are the corresponding standard deviations
and sY Yb represents the covariance. Therefore, MSEP can be expressed as
MSEP =
1
t
2
2
2
b
b
y
−
y
+ sY − sYb + 2 sY sYb − sY Yb = EB + EV + ER ,
(y
−
y
)
=
∑ n+ j n+ j
t
j=1
or, equivalently, with the identity
1=
EB
EV
ER
+
+
= UB +UV +UR ,
MSEP MSEP MSEP
where UB is the part of MSEP corresponding to the bias due to the systematic prediction
errors; UV indicates the difference between the variability of the real values and the
variability of the predicted values; finally, UR shows the random variability in the
prediction errors.
Marı́a Isabel Sánchez-Rodrı́guez et al.
237
A model is obviously better as MSEP approaches 0 (taking into account that MSEP
is not upper bounded and depends on the unit of measurement). But, using the proposed
decomposition, if MSEP shows a great percentage attributable to systematic errors,
this aspect indicates that there is some detectable cause causing these deviations in the
predictions. This cause must be detected in order to eliminate systematic errors. Thus, a
great percentage of MSEP attributable to systematic prediction errors indicates that the
model can be improved in some sense. Nevertheless, this improvement is difficult if the
predictions generated by a model have a random nature because random errors, with a
white noise appearance, are usually inherent to a process.
Definitively, the ideal situation for evaluating the predictive capability of a model is
presented when MSEP has a value as close as possible to 0 and besides UB = 0, that
is, systematic errors do not exist in the prediction; UV = 0, which indicates that the
variability of the real values is the same as that of the predictions; and UR = 1, which
corresponds to prediction errors with random nature.
3.2. New methodological approaches in the chemometric analysis
of IR spectra
Now, the procedures used in the literature to extract information of olive oil from IR
spectra (NIR, MIR and concatenated NIR-MIR) are revised. The different contributions
to each technique are conveniently motivated and justified.
1. Extraction of the information from the complete IR spectra versus the analysis of some particular IR bands. There are in the literature many references in
which the analysis of IR spectra is made based on the detection of highly informative bands. One such example is the work by Guillén and Cabo (1997), who
relate IR spectral bands of edible oils with some chemical functional groups. This
approach is based on the Lambert-Beer Law, which states that the intensities of the
spectral bands are proportional to the concentration of their respective functional
groups. The frequencies of some bands, fundamentally the ones associated to the
so-called fingerprint region1 , are highly correlated to the composition of olive oil.
Guillén and Cabo (1997) successfully obtained regression equations to predict the
content in SAFA, MUFA and PUFA fatty acids of olive oil from the frequencies
of some bands in the fingerprint region (see Table 1). A follow-up study (Guillén
and Cabo, 1998) generalized the previous results by regression models that provide relationships between the composition in SAFA, MUFA and PUFA of edible
oils and the ratio of absorbance of specific bands of the IR spectra, not necessarily
1. The region 1500-700 cm−1 of the MIR spectra is named fingerprint region because this region is highly characteristic of a specific compound. Little changes in the molecular structure frequently cause significant changes in the
absorption peaks of this region.
238
New approaches in the chemometric analysis of infrared spectra of extra-virgin olive oils
associated to the fingerprint region. Besides, Guillén and Cabo (1999) used the
previous regression equations to determine the composition of mixtures of olive
oil and other low quality oils (such as sunflower or peanut), using gas chromatography values as references.
There are other works which analyse IR spectra by determining relevant frequency
bands. Vlachos et al. (2006) establish the relation between the frequency 3009
cm−1 of the IR spectra and the percentage of adulteration of olive oil with low
quality oils. Rohman and Man (2010) use PCA and PLS components extracted
from the fingerprint region 1500-1000 cm−1 (MIR spectra) to quantitatively and
qualitatively analyse extra-virgin olive oils, to detect possible adulteration with
palm oil. Nicoletta et al. (2010) select some regions from NIR and MIR spectra to
classify, by discriminant analysis, diverse extra-virgin olive oils based on their
fruity attribute intensity. Zhang et al. (2012) divide the IR spectra in regions,
attending to the absorbance peaks, to establish linear regression equations to detect
possible adulteration of vegetables oils with used frying oils.
All the previously cited works determine, by using correlation, highly informative
IR spectral regions to predict the composition of olive oil. In general, the determined zones are localized in the mid infrared spectral region (MIR), where the
spectral fingerprint is localized.
Our previous study (Sanchez-Rodriguez et al., 2013), from NIR spectral data,
compares the estimation results obtained by extracting information from the whole
spectra with those provided by some specific NIR bands (either determined by
cluster analysis or associated to certain spectral peaks). The best calibration and
validation results are obtained from the whole spectra. This is the reason why
the present work uses the whole NIR, MIR and concatenated NIR-MIR spectra to
Table 1: Coefficients for IR equations, Frequency = a + b%M (%P o %S),
and linear correlation coefficientsa (Guillén and Cabo, 1997).
a
Percentage
a
b(10−2 )
r
M
P
M
P
M
P
S
S
S
S
S
S
3010.40
3004.85
1394.90
1402.61
1100.46
1096.68
2926.04
2855.07
1122.65
724.30
1238.11
1160.20
-7.24
+6.10
+9.90
-8.43
-4.87
+4.43
-6.28
-6.59
-22.65
-9.93
+2.33
24.44
0.9853
0.9492
0.9910
0.9223
0.9908
0.9176
0.8504
0.9565
0.9510
0.9924
0.8408
0.9989
%M, %P and %S represents the percentage of MUFA, PUFA and SAFA, respectively.
Marı́a Isabel Sánchez-Rodrı́guez et al.
239
predict the content of olive oil in SAFA, MUFA and PUFA fatty acids. As subsequently shown, although in the analysis of spectral bands the most informative
zones are localized in the MIR spectral region, the NIR spectra provides better
estimations in certain situations when the whole spectral information is used.
2. PLS regression versus general linear regression or PCA regression. The analysis of IR spectra from the detection of relevant wavelength bands to obtain quantitative information (such as the prediction of the content of olive oil in some specific
compounds) is based on the matching of some wavelengths with high correlation
with the response variable. Then linear regression equations are fit to predict the
percentage of the compound as a function of the wavelengths (see, for example,
Guillén and Cabo (1997, 1998, 1999), Vlachos et al. (2006), Rohman and Che
Man (2010), Zhang et al. (2012)). The selection of a single wavelength could lead
to a waste of useful statistical information. But the selection of many wavelengths
highly correlated with the dependent variable could cause the presence of multicollinearity among the explanatory variables, incompatible with the hypothesis of
uncorrelation in the general linear model. This is why the use of principal component regression (PCA regression) or partial least squares regression (PLS regression) is more interesting. Both methodologies summarize the information of the
whole IR spectrum in some latent factors or components, orthogonal among themselves, thus avoiding the possible multicollinearity in the model. These factors are
obtained as linear combinations of the independent variables in both methodologies. However, the factors are obtained by maximizing the covariances (or correlations) among the explanatory variables, in PCA regression, and the covariances
(or correlations) between the explanatory variables and the dependent one, in PLS
regression.
In this work, PLS regression has been selected because a previous one (SánchezRodrı́guez et al. (2013)) highlights the benefits of PLS regression versus PCA
regression in the determination of quantitative information from NIR data. There
are also other works indicating that PLS regression is better than PCA regression
in the multivariate analysis of NIR or MIR spectral data (see, for example, Frank
and Friedman (1993) or Maggio et al. (2011)).
3. Progressive introduction of PLS factors in the regression model. The estimation algorithms that compare the MSEC and MSEP values obtained for PCA and
PLS regression or for PCA or PLS discriminant analysis (PCA-DA or PLS-DA)
have established, in the chemometric software, a stopping internal criterion to determine the number of factors to retain. This is the case, for example, of the article by Dupuy et al. (2010a) and (2010b), which uses UNSCRAMBLER software
version 9.8 from CAMO (Computer Aided Modelling, Trondheim, Norway) and
Matlab software from MathWorks in the analysis of NIR, MIR and concatenated
NIR-MIR spectra.
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New approaches in the chemometric analysis of infrared spectra of extra-virgin olive oils
The criteria determining the number of factors to retain in PLS regression are
diverse. For example, in PCA, the Kaiser criterion is the default in most statistical
software. It suggests that those principal components with eigenvalues equal to
or higher than 1 should be retained, as each eigenvalue represents the variance of
the corresponding factor. In PLS analysis, the criterion of the first increase of the
mean squared error of prediction is considered: the number of latent factors taken
into account is
h∗ = min{h > 1 : MSEP(h + 1) − MSEP(h) > 0},
where MSEP(h) is the mean squared error of prediction of the regression model
with h factors. Gowen et al. (2010) present some measures for preventing the overfitting in PLS calibration models of near-infrared (NIR) spectroscopy data and
investigate the simultaneous use of both model bias and variance in the selection
of the number of latent factors to include in the model.
The cited criteria have an empirical character and are not unanimously applied.
Therefore, the present work does not fix the number of factors. This number is
progressively incremented in the PLS regression model and the associated MSEC
and MSEP values are compared. As subsequently confirmed, the NIR matrix of
data provides better estimations of the content in fatty acids of olive oil than MIR
matrix for a lower number of factors whereas the opposite holds true for a higher
number of latent factors.
4. Procedures of cross-validation. The procedures of cross-validation are aimed at
avoiding the bias associated to the case of using the same data to fit the regression
model and to evaluate the corresponding predictions. They are repetitive algorithms that, at each iteration, subdivide the set of original data in the calibration
and the validation subsets. The calibration (or fit) set, formed by 80% of the data
approximately, is used in the fit of the model and provides the MSEC value as a
measure of goodness-of-fit. But the validation (or prediction) set, formed by the
remaining 20% of the data, is reserved in the training of the model and so can be
used to evaluate its predictive capability with the MSEP value.
The computer programs frequently used in Chemometrics have some cross-validation procedures implemented. They iteratively repeat the selection of calibration
and validation sets and provide and average of the MSEC and MSEP values obtained for each iteration. This is the case, for example, of the paper by Rohman
and Man (2010), that uses the software TQ AnalystTM Version 6 (Thermo Electron
Corporation, Madison, WI); Sinelli et al. (2010), which uses the V-PARVUS package (Forina et al. (2008)); Dupuy et al. (2010b), that uses the UNSCRAMBLER
software version 9.8 from CAMO.
Marı́a Isabel Sánchez-Rodrı́guez et al.
241
But the present work calculates and represents the MSEC and MSEP values
obtained for different random selections of the calibration and validation sets, from
the NIR, MIR and concatenated NIR-MIR matrices of data. The algorithm has
been programmed by using the Matlab software from MathWorks. The graphical
representations permit to compare not only the mean MSEC and MSEP values but
also their variability in the successive selections. The three-dimensional graphics
permit to compare the results for the three type of acids and matrices of data
simultaneously.
5. Decomposition of the mean squared error of prediction. With the aim of
analyzing the nature of the prediction errors, this work uses a decomposition of
the MSEP value in the terms EB , EV and ER , attributable to systematic errors, the
difference in variability among the real and the predicted values and random errors,
respectively (Section 3.1, Sánchez-Rodrı́guez et al. (2013)). This decomposition
is presented for the predictions for each type of fatty acid (SAFA, MUFA and
PUFA) and spectral zone (NIR, MIR and concatenated NIR-MIR), as a function
of the number of PLS factors in the regression model. Besides, in the context of
cross-validation, this decomposition is also presented for the successive selections
of calibration and validation sets. The ideal situation for evaluating the predictive
capability of a model is presented when MSEP has a value nearer to 0 and the
great percentage of this value is associated to the randomness and the lowest
percentage is attributable to systematic errors. This work afterwards highlights
that these percentages depend on the type of fatty acid to estimate, the IR spectral
zone used for the estimation (NIR, MIR or NIR-MIR) and the number of PLS
factors in the regression model.
6. Treatment of the spectra in the context of functional data analysis. A line of
future research (see Sánchez-Rodrı́guez and Caridad (2014)) could consider IR
spectra as so-called data objects in object-oriented data analysis (OODA). This is
the particular case of functional data analysis (FDA), in which the data objects of
OODA are curves (see the overview by Marron and Alonso (2014)). In this context, multivariate techniques such as PCA or PLS regression, have been extended
to the functional case. For example, Aguilera et al. (2010) apply functional PLS
and PCA regressions to simulated and spectrometric data, comparing the results
with the corresponding discrete ones and concluding that functional PLS regression provides better estimations of the parameter function than functional PCA
regression and similar predictions. Preda and Saporta (2005) apply functional regression models to predict the behaviour of shares and conclude that the functional
PLS regression model provides the best forecasts evaluating the global model quality by the sum of squared errors. Finally, also the classification techniques, such as
logit regression or discriminant analysis, have been successfully extended to the
functional case (see, respectively, Escabias et al. (2007) or Preda et al. (2007)).
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New approaches in the chemometric analysis of infrared spectra of extra-virgin olive oils
4. Results and discussion
Initially, Section 4.1 deals with the estimation of SAFA, MUFA and PUFA fatty acids of
olive oils by PLS regression from the NIR, MIR and concatenated NIR-MIR matrices of
data. The results of calibration and validation depend on the number of PLS factors in
the regression model. These results are compared for the three matrices of data and types
of fatty acids. The randomness of the prediction errors is analysed by a decomposition of
MSEP. Subsequently, in Section 4.2, the previous results are generalized by using crossvalidation procedures, that is, changing iteratively the training and the test data sets.
Besides, three-dimensional scatterplots permit to obtain conclusion simultaneously for
the three matrices of data or types of fatty acids.
4.1. Chemometrics from IR data: progressive introduction of PLS factors
in the regression model
NIR and MIR spectroscopies provide n × p data matrices whose rows refer to an
olive oil (n = 128, in total) and each column is associated to a wavelength of the
spectrum (pNIR = 1237 and pMIR = 1843). The information given by NIR and MIR
data is summarized by using PLS regression. Sánchez-Rodrı́guez et al. (2013) pointed
out that this technique, applied directly to the matrix of NIR data, provides a potential
methodology to predict the content in fatty acids of olive oil. This paper shows that the
results obtained from the whole matrix of data, being considered as a “black box”, are
better than the ones obtained with the selection of some spectral peaks or spectral regions
by cluster analysis. Besides, PLS regression considerably improves, in this context, the
results obtained for PCA regression. As stated above, both PCA and PLS methodologies
provide components or factors orthogonal among themselves, thus avoiding the possible
presence of multicollinearity in the regression model.
With regard to the rows of the NIR and MIR data matrices, 80% of them, randomly
selected, will be used for calibration and the remaining 20%, for prediction or validation.
Initially, the NIR matrix of data is considered and PLS components are extracted. Those
components will be progressively introduced in the PLS regression models that consider
the content in SAFA, MUFA and PUFA fatty acids as explained variables, respectively.
For each number of introduced components, the mean squared error of calibration and
prediction, MSEC and MSEP, are calculated. The same process is repeated considering,
secondly, the MIR data matrix and, finally, the concatenated NIR-MIR data matrix. In
addition, with the purpose of determining the character, systematic or random, of the
prediction errors, a decomposition of MSEP obtained by the PLS regression models on
NIR, MIR and NIR-MIR matrices (in the UB , UV and UR components) is obtained for
each fatty acid.
Marı́a Isabel Sánchez-Rodrı́guez et al.
243
Figures 3, 4 and 5: MSEC in the estimation of SAFA, MUFA and PUFA from PLS components
of NIR, MIR and concatenated NIR-MIR matrices.
Figures 6, 7 and 8: MSEP in the estimation of SAFA, MUFA and PUFA from PLS components
of NIR, MIR and concatenated NIR-MIR matrices.
Figures 3, 4 and 5 represent the MSEC in the estimation of SAFA, MUFA and PUFA
acids obtained by successively introducing components in the regression model. These
components are obtained on the NIR, MIR and concatenated NIR-MIR matrices of data.
These figures show that, for the three types of fatty acids, the NIR (and concatenated
NIR-MIR) matrix of data provides better calibration results in regression models with
a lower number of PLS factors. But MIR (and concatenated NIR-MIR) matrix supplies
better estimations for models with a higher number of factors.
Then, Figures 6, 7 and 8 represent, in the same context, the respective MSEP
values. MSEP evaluates the predictive capability of a model, taking into account that
the estimations are calculated by using observations that are not included in the fit or
calibration of the model. The conclusions in prediction are similar to the ones obtained in
calibration: the MSEP values obtained from MIR data are lower than the ones obtained
from NIR data when the number of PLS factors in the model is sufficiently high, but not
for low values.
Figures 9, 10 and 11 show the UR term in the decomposition of MSEP for SAFA,
MUFA and PUFA acids, respectively. This term corresponds to random prediction errors
and, as in the previous graphics, is expressed as a function of the number of PLS
components in the model. The figures evidence that the UR term represents the great
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New approaches in the chemometric analysis of infrared spectra of extra-virgin olive oils
Figures 9, 10 and 11: UR term of MSEP in the estimation of SAFA, MUFA and PUFA
from PLS components of NIR, MIR and concatenated NIR-MIR matrices.
percentage for each case, as this ratio is near to 1. With respect to the comparison of
the techniques, there are differences depending on the used NIR, MIR and concatenated
NIR-MIR matrices of data and the estimated fatty acid. For a higher number of PLS
factors in the model, the three NIR, MIR and NIR-MIR UR terms are very close to one.
But for a lower number of factors, the MIR UR term is closer to one in the estimation
of SAFA and PUFA acids but it is farther from one in the estimation of MUFA acids. In
this last case, the NIR matrix of data provides better results.
These results suggest that, under our experimental conditions, a more accurate
estimation in calibration and validation of SAFA, MUFA and PUFA content in extravirgin olive oil (taking GC as the reference technique) is obtained from the NIR matrix
for a lower number of PLS factors. For a greater number of PLS factors, the MIR
matrix provides the best results. The previous considerations are important as usual
chemometric computer programs have internally implemented a stopping criterion to
retain a concrete number of PLS factors in the regression model. It is interesting to
identify the range of variation of this number to determine the region of the IR spectra,
NIR or MIR, that provides better estimations of the different fatty acids. Then, analysing
the nature of the prediction errors, the percentage of them attributable to random
causes also depends on the region of the IR spectra and the type of acid. The MIR
matrix provides, in general, better results in the estimation of SAFA and PUFA acids,
irrespective of the number of PLS factors in the model. On the contrary, in the estimation
of MUFA acids, NIR matrix supplies better results, also independently of the number of
factors.
4.2. Cross-validation: generalization of the previous results
In the last subsection, the original data have been subdivided in a single calibration
set (containing the 80% of the original data, specifically, 102 out of 128 data) and a
single validation set (with the 20%, that is, 26 data). The calibration set is used to train
the regression model. The validation set is used to test the model, using data reserved
in the fit of the model. With the goal of generalizing the previously obtained results,
Marı́a Isabel Sánchez-Rodrı́guez et al.
245
procedures of cross-validation are used in this section. They are implemented by a
repetitive algorithm that, for each iteration, modifies the partition in calibration and
validation subsets of the original data set. For each iteration, MSEC and MSEP are
calculated for evaluating, respectively, the goodness-of-fit and the predictive capability
of the corresponding model.
More specifically, the cross-validation algorithm has been implemented for 30 iterations, randomly selecting, for each one, the sets considered for calibration and validation. Besides, since the previous section highlights differences depending on the number
of PLS factors in the regression model, this section compares the results for a low number of factors, 5, and also for a high number of factors, 30.
Figures 12 and 13 draw three-dimensional scatterplots for goodness-of-fit or calibration in cross-validation. The point clouds are associated to models with 5 and 30
PLS factors, respectively, representing MSECSAFA , MSECMUFA and MSECPUFA in x, y,
z axes. Unlike the previously represented figures, these graphics permit to compare, in a
global manner, the results obtained for the three types of fatty acids simultaneously. For
5 PLS factors (Figure 12), the MIR point cloud is farther from the origin (0,0,0) than
the corresponding to the NIR (and NIR-MIR) data. For 30 PLS factors (Figure 13), the
conclusions are the opposite: in this case, the estimations from NIR data are associated
with the high MSEC values.
Figures 12 and 13: MSEC obtained by the cross-validation algorithm from NIR, MIR
and concatenated NIR-MIR data (for SAFA, MUFA and PUFA) for 5 and 30 factors, respectively.
The same conclusions are obtained for the three fatty acid types, if the models
are compared in validation or prediction terms (using MSEP, see Figures 14 and 15).
Besides, the variability existing among the MSEP values is higher for the MIR than for
the NIR estimations in the models with 5 PLS factors and lower for the models with 30
PLS factors.
Figures 16-18 represent, for each acid type, the decomposition of MSEP in the
terms UB , UV and UR obtained, by the cross-validation algorithm, for each iteration. The
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New approaches in the chemometric analysis of infrared spectra of extra-virgin olive oils
Figures 14 and 15: MSEP obtained by the cross-validation algorithm from NIR, MIR
and concatenated NIR-MIR data (for SAFA, MUFA and PUFA) for 5 and 30 factors, respectively.
Figures 16, 17 and 18: Decomposition of MSEP obtained by the cross-validation algorithm
in the estimation of SAFA, MUFA and PUFA from PLS components of NIR,
MIR and concatenated NIR-MIR matrices, respectively.
aim is to determine the nature, random or systematic, of the prediction errors. The point
clouds represent the values corresponding to NIR, MIR and NIR-MIR matrices in x, y,
z axes, respectively. It is evident that, for each case, the component corresponding to
random prediction errors, UR , is associated to the great percentage, as this ratio is near
to 1 for NIR, MIR and NIR-MIR axes. This is the suitable situation in the evaluation of
the predictive character of a model.
Finally, with the aim to confirm the differences detected previously depending on
the number of the PLS factors in the model, Figures 19 and 20 depict the UR term
associated with the models with 5 and 30 factors. The scatterplots represent the values
corresponding to UR, SAFA , UR, MUFA and UR, PUFA in x, y, z axes, respectively. The results
show that the UR term obtained from the NIR, MIR and concatenated NIR-MIR matrices
of data is close to 1 in models with a relatively high number of factors (30). But, in
model with a low number of factors, the NIR and NIR-MIR UR terms are lower, clearly
discriminated from the one obtained from the MIR data.
Marı́a Isabel Sánchez-Rodrı́guez et al.
247
Figures 19 and 20: UR term of MSEP obtained by the cross-validation algorithm from NIR, MIR
and concatenated NIR-MIR data (for SAFA, MUFA and PUFA) for 5 and 30 factors, respectively.
5. Conclusions
In recent years, procedures which permit to determine in a fast and efficient manner the
profile of olive oils in different components have been generalized, specially aiming at
evaluating quality indexes. In this sense, spectroscopic techniques have been extended.
In parallel, multivariate statistics has emerged as a powerful tool to identify and extract
the information contained in spectra.
In this work, Chemometrics is applied to data obtained from IR spectra, in the near
(NIR) and mid (MIR) zones, and using GC data as a reference. PLS regression models
to predict the content in SAFA, MUFA and PUFA fatty acids of olive oil are proposed,
using the three NIR, MIR and concatenated NIR-MIR matrices of data. The final
conclusion is that the best estimation of calibration or fit and validation or prediction are
obtained from the NIR data for lower numbers of PLS factors and from the MIR data
for higher numbers of factors. This is important to be taken into account since, usually,
chemometric computer programs have a stopping criterion implemented to determine
the number of PLS factors to be retained.
These conclusions are generalized via cross-validation procedures. They compare
estimations in terms of goodness-of-fit and prediction for different calibration and
validation subsets and evidence the desirable main random nature of the estimation
errors. Three-dimensional scatterplots confirm the differences among the three fatty acid
types and matrices simultaneously.
Then, this study analyses the prediction errors to determine their nature, systematic
or random. Also in this case the conclusions depend on the number of PLS latent factors,
the type of fatty acid and the matrix of data. These differences are detected by the UR
term, that represents the percentage of randomness in the prediction errors. In general,
irrespective of the number of factors in the regression model, the MIR zone provides a
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New approaches in the chemometric analysis of infrared spectra of extra-virgin olive oils
higher value in the estimation of SAFA and PUFA acids. But, in the estimation of MUFA
acids, the NIR matrix gives better estimations. In the three-dimensional representation
of the UR term for the three acids and IR zones, this term is always close to 1 for a high
number of PLS factors. But, for a low number of factors, the NIR and NIR-MIR UR
terms are clearly lower than the associated to the MIR data.
Acknowledgements
The authors thank the financial support by ‘Junta de Andalucı́a’ (Project P08-FQM3931) and FEDER funds.
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